Post 3: Definite Integrals and Area Calculation

In the previous post, we discussed approximating the area under a curve using Riemann sums. While Riemann sums provide a good approximation, we can obtain the exact area under a curve using definite integrals. Definite integrals are a powerful tool in calculus for finding the area of a region bounded by a curve. Let's dive deeper into this concept.

Definite Integrals A definite integral is a mathematical concept representing the exact area between a function and the x-axis over a given interval. It allows us to calculate the area under a curve precisely. We denote a definite integral using the symbol ∫.

The definite integral of a function f(x) over an interval [a, b] is expressed as:

∫[a, b] f(x) dx

Here, f(x) represents the function, dx denotes an infinitesimally small change in x, and [a, b] defines the interval over which the area is calculated.

Evaluating Definite Integrals To evaluate definite integrals, we need to find an antiderivative of the function and use the Fundamental Theorem of Calculus. The antiderivative of f(x) is denoted as F(x). The Fundamental Theorem of Calculus states:

∫[a, b] f(x) dx = F(b) - F(a)

In simple terms, we find the antiderivative of f(x) and substitute the limits of integration, a and b, into this antiderivative. Then, we subtract the antiderivative value at a from the antiderivative value at b to obtain the area.

Example: Let's find the area between the curve f(x) = 2x + 3 and the x-axis over the interval [1, 5].

Step 1: Find the antiderivative of f(x): The antiderivative of f(x) = 2x + 3 is F(x) = x^2 + 3x + C, where C is the constant of integration.

Step 2: Evaluate the definite integral: ∫[1, 5] (2x + 3) dx = [x^2 + 3x] from 1 to 5 = [(5^2 + 3(5))] - [(1^2 + 3(1))] = [25 + 15] - [1 + 3] = 40 - 4 = 36

Therefore, the area between the curve f(x) = 2x + 3 and the x-axis over the interval [1, 5] is 36 square units.

Definite integrals help us find the exact area under more complex curves, which is difficult to achieve using other methods. It is a crucial concept in calculus that has numerous applications in various fields of study, as we'll explore in the next post.